Open Access
October 2018 Smooth backfitting for errors-in-variables additive models
Kyunghee Han, Byeong U. Park
Ann. Statist. 46(5): 2216-2250 (October 2018). DOI: 10.1214/17-AOS1617


In this work, we develop a new smooth backfitting method and theory for estimating additive nonparametric regression models when the covariates are contaminated by measurement errors. For this, we devise a new kernel function that suitably deconvolutes the bias due to measurement errors as well as renders a projection interpretation to the resulting estimator in the space of additive functions. The deconvolution property and the projection interpretation are essential for a successful solution of the problem. We prove that the method based on the new kernel weighting scheme achieves the optimal rate of convergence in one-dimensional deconvolution problems when the smoothness of measurement error distribution is less than a threshold value. We find that the speed of convergence is slower than the univariate rate when the smoothness of measurement error distribution is above the threshold, but it is still much faster than the optimal rate in multivariate deconvolution problems. The theory requires a deliberate analysis of the nonnegligible effects of measurement errors being propagated to other additive components through backfitting operation. We present the finite sample performance of the deconvolution smooth backfitting estimators that confirms our theoretical findings.


Download Citation

Kyunghee Han. Byeong U. Park. "Smooth backfitting for errors-in-variables additive models." Ann. Statist. 46 (5) 2216 - 2250, October 2018.


Received: 1 November 2016; Revised: 1 July 2017; Published: October 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06964331
MathSciNet: MR3845016
Digital Object Identifier: 10.1214/17-AOS1617

Primary: 62G08
Secondary: 62G20

Keywords: Deconvolution , errors-in-variables models , kernel smoothing , Nonparametric additive regression , smooth backfitting

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 5 • October 2018
Back to Top